3.80.33 \(\int \frac {-3+x-2 x \log (x)+x \log (x) \log (\log (x))}{x \log (x)} \, dx\)

Optimal. Leaf size=15 \[ 1+x-(-3+x) (3-\log (\log (x))) \]

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Rubi [A]  time = 0.11, antiderivative size = 14, normalized size of antiderivative = 0.93, number of steps used = 9, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {6688, 2353, 2298, 2302, 29, 2520} \begin {gather*} -2 x+x \log (\log (x))-3 \log (\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-3 + x - 2*x*Log[x] + x*Log[x]*Log[Log[x]])/(x*Log[x]),x]

[Out]

-2*x - 3*Log[Log[x]] + x*Log[Log[x]]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 2298

Int[Log[(c_.)*(x_)]^(-1), x_Symbol] :> Simp[LogIntegral[c*x]/c, x] /; FreeQ[c, x]

Rule 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 2353

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol]
:> With[{u = ExpandIntegrand[(a + b*Log[c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[
{a, b, c, d, e, f, m, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[m] && IntegerQ[r
]))

Rule 2520

Int[Log[Log[(d_.)*(x_)^(n_.)]^(p_.)*(c_.)], x_Symbol] :> Simp[x*Log[c*Log[d*x^n]^p], x] - Dist[n*p, Int[1/Log[
d*x^n], x], x] /; FreeQ[{c, d, n, p}, x]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-2+\frac {-3+x}{x \log (x)}+\log (\log (x))\right ) \, dx\\ &=-2 x+\int \frac {-3+x}{x \log (x)} \, dx+\int \log (\log (x)) \, dx\\ &=-2 x+x \log (\log (x))+\int \left (\frac {1}{\log (x)}-\frac {3}{x \log (x)}\right ) \, dx-\int \frac {1}{\log (x)} \, dx\\ &=-2 x+x \log (\log (x))-\text {li}(x)-3 \int \frac {1}{x \log (x)} \, dx+\int \frac {1}{\log (x)} \, dx\\ &=-2 x+x \log (\log (x))-3 \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log (x)\right )\\ &=-2 x-3 \log (\log (x))+x \log (\log (x))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 14, normalized size = 0.93 \begin {gather*} -2 x-3 \log (\log (x))+x \log (\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3 + x - 2*x*Log[x] + x*Log[x]*Log[Log[x]])/(x*Log[x]),x]

[Out]

-2*x - 3*Log[Log[x]] + x*Log[Log[x]]

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fricas [A]  time = 0.62, size = 11, normalized size = 0.73 \begin {gather*} {\left (x - 3\right )} \log \left (\log \relax (x)\right ) - 2 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x*log(x)*log(log(x))-2*x*log(x)+x-3)/x/log(x),x, algorithm="fricas")

[Out]

(x - 3)*log(log(x)) - 2*x

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giac [A]  time = 0.25, size = 14, normalized size = 0.93 \begin {gather*} x \log \left (\log \relax (x)\right ) - 2 \, x - 3 \, \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x*log(x)*log(log(x))-2*x*log(x)+x-3)/x/log(x),x, algorithm="giac")

[Out]

x*log(log(x)) - 2*x - 3*log(log(x))

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maple [A]  time = 0.03, size = 15, normalized size = 1.00

method	result	size
default	\(-2 x -3 \ln \left (\ln \relax (x )\right )+x \ln \left (\ln \relax (x )\right )\)	\(15\)
norman	\(-2 x -3 \ln \left (\ln \relax (x )\right )+x \ln \left (\ln \relax (x )\right )\)	\(15\)
risch	\(-2 x -3 \ln \left (\ln \relax (x )\right )+x \ln \left (\ln \relax (x )\right )\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*ln(x)*ln(ln(x))-2*x*ln(x)+x-3)/x/ln(x),x,method=_RETURNVERBOSE)

[Out]

-2*x-3*ln(ln(x))+x*ln(ln(x))

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maxima [A]  time = 0.45, size = 14, normalized size = 0.93 \begin {gather*} x \log \left (\log \relax (x)\right ) - 2 \, x - 3 \, \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x*log(x)*log(log(x))-2*x*log(x)+x-3)/x/log(x),x, algorithm="maxima")

[Out]

x*log(log(x)) - 2*x - 3*log(log(x))

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mupad [B]  time = 4.83, size = 14, normalized size = 0.93 \begin {gather*} x\,\ln \left (\ln \relax (x)\right )-3\,\ln \left (\ln \relax (x)\right )-2\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x - 2*x*log(x) + x*log(log(x))*log(x) - 3)/(x*log(x)),x)

[Out]

x*log(log(x)) - 3*log(log(x)) - 2*x

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sympy [A]  time = 0.31, size = 15, normalized size = 1.00 \begin {gather*} x \log {\left (\log {\relax (x )} \right )} - 2 x - 3 \log {\left (\log {\relax (x )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x*ln(x)*ln(ln(x))-2*x*ln(x)+x-3)/x/ln(x),x)

[Out]

x*log(log(x)) - 2*x - 3*log(log(x))

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